"Please check your luggage."
请你检查行李 is grammatically correct and I am quite sure you would be understood in a real life situation.
It is preferred to leave things implied in Chinese but only as long as there is little chance of creating confusion. When you pull away the attributive 你的 from the noun 行李, you cannot convey the specific meaning of whose luggage it is. Then you may be misunderstood as meaning luggage in general. Whether you can make this omission depends on the context.
一下 can be omitted too, but keeping it effectively makes the request softer and friendly.
You can, but sometimes it is not about the speed at all. e.g. 请您等一下can hardly be translated as Please wait quickly.
In most of the time, 一下 is referring to an action not being done thoroughly (without using full power, without paying full attention, without lasting a long time, without a large number of repetition, without doing it on a large number of objects, etc.) A (little) bit is often a good translation.
Hi Keith, thanks for sharing your thoughts. It could be a regional difference but here in the UK it's actually really common to say something like, "Please wait here quickly”. It refers not to the speed of the action itself but rather that it is of short duration. In terms of the example above about luggage, I just think it sounds a bit strange to say, "Please check your luggage a little bit.”
Ah, perhaps not quite so strong a softener. It is merely to make it sound friendlier, 请 makes it sufficiently polite, as it were, already. I don't have anything in mind for that right now, everything I think of seems to verge into a question, or just the tone of voice and body language accompanying the sentence itself.